Congruence primes for Ikeda lifts and the Ikeda ideal

Jim Brown and Rodney Keaton

Vol. 274 (2015), No. 1, 27–52

DOI: 10.2140/pjm.2015.274.27

Abstract

Let $f$ be a newform of level 1 and weight $\left(2\kappa -n\right)$ for positive even integers $\kappa$ and $n$. We study congruence primes for the Ikeda lift of $f$. In particular, we consider a conjecture of Katsurada stating that primes dividing certain $L$-values of $f$ are congruence primes for the Ikeda lift. Instead of focusing on a congruence to a single eigenform, we deduce a lower bound on the number of all congruences between the Ikeda lift of $f$ and forms not lying in the space spanned by Ikeda lifts.

Keywords

Ikeda lifts, Siegel modular forms, congruences between modular forms

Mathematical Subject Classification 2010

Primary: 11F33

Secondary: 11F46, 11F30, 11F55

Milestones

Received: 16 October 2013

Revised: 20 January 2014

Accepted: 24 January 2014

Published: 2 March 2015